Problem: In the diagram, $RSP$ is a straight line and $\angle QSP = 80^\circ$.  What is the measure of $\angle PQR$, in degrees?

[asy]
draw((.48,-.05)--(.48,.05)); draw((.52,-.05)--(.52,.05)); draw((1.48,-.05)--(1.48,.05)); draw((1.52,-.05)--(1.52,.05));

draw((1.04,.51)--(1.14,.49)); draw((1.03,.47)--(1.13,.45));
draw((0,0)--(2,0)--(1.17,.98)--cycle);
label("$P$",(2,0),SE); label("$R$",(0,0),SW); label("$Q$",(1.17,.98),N);

label("$80^\circ$",(1,0),NE);

label("$S$",(1,0),S);
draw((1,0)--(1.17,.98));
[/asy]
Solution: Since $RSP$ is a straight line, we have $\angle RSQ+\angle QSP = 180^\circ$, so $\angle RSQ=180^\circ - 80^\circ = 100^\circ$. $\triangle RSQ$ is isosceles with $RS=SQ$, so \[ \angle RQS = \frac{1}{2}(180^\circ - \angle RSQ) = \frac{1}{2}(180^\circ - 100^\circ)=40^\circ . \]Similarly, since $\triangle PSQ$ is isosceles with $PS=SQ$, we have \[ \angle PQS = \frac{1}{2}(180^\circ - \angle PSQ) = \frac{1}{2}(180^\circ - 80^\circ)=50^\circ . \]Therefore, $\angle PQR = \angle PQS + \angle RQS = 50^\circ+40^\circ=\boxed{90}^\circ$.